Determine whether each equation defines y as a function of x. x = (1/3)(y²)

Verified step by step guidance

1. Start by examining the given equation: \(x = \frac{1}{3} y^{2}\). This equation relates \(x\) and \(y\).
2. To determine if \(y\) is a function of \(x\), try to express \(y\) explicitly in terms of \(x\). Multiply both sides by 3 to isolate \(y^{2}\): \$3x = y^{2}$.
3. Next, solve for \(y\) by taking the square root of both sides: \(y = \pm \sqrt{3x}\). Notice the \(\pm\) sign indicates two possible values of \(y\) for each \(x\) (except when \(x=0\)).
4. Recall the definition of a function: for each input \(x\), there must be exactly one output \(y\). Since here each \(x\) (greater than or equal to zero) corresponds to two \(y\) values, \(y\) is not a function of \(x\).
5. Therefore, the equation does not define \(y\) as a function of \(x\) because it fails the vertical line test and does not assign a unique \(y\) to each \(x\).

Key Concepts

Definition of a Function

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if an equation defines y as a function of x, we check if for every x there is only one y.

Solving for y in Terms of x

To analyze if y is a function of x, we often solve the equation for y explicitly. This helps identify if multiple y-values correspond to a single x-value, which would mean y is not a function of x.

Vertical Line Test and Relation Symmetry

The vertical line test visually checks if a curve represents y as a function of x by ensuring vertical lines intersect the graph at most once. Equations involving y² often produce symmetric graphs, indicating multiple y-values for one x.